On combining equations (10.99)and (10.100), we get
with
where
is generalized force vector, the mass matrix 
consists of the translational mass matrix
and the rotational mass matrix 
. Details of the mass matrix and stiffness matrices are given in Appendix 10.1. Now, through examples the effect of rotary inertia and shear deformation would be illustrated.
Example 10.2 Obtain n atural frequency parameters defined by a non-dimensional form 
as of a uniform, non-rotating simply supported Timoshenko beam for the first four modes. The slenderness parameter R(=r/2L) is to be varied from 0.02 (thin beam) to 0.10 (thick beam). Show the comparison for different number of elements and with the exact analytical formula.
Solution : Since in previous chapters the usual finite element procedures have been dealt in detail (i.e., regarding the elemental equations, assembly procedures, application of boundary conditions, eigen value problem formulations, etc.), hence here those details for the present problem is omitted. The uniform, non-rotating simply supported Timoshenko beam for obtaining natural frequencies, is first discretised into five numbers of elements. From the eigen value formulation of the present problem first four lowest natural frequencies obtained and are compared with the published work (Ku 1998) and by the classical closed form solution given by Shames and Dym,2005.Ku (1998) has considered C0 continuity (i.e., compatibility up to translational displacements) for the Timoshenko beam model whereas model represented here considers C1 continuity (i.e., the compatibility up to the translational and rotational displacements). Comparisons of non-dimensional natural frequencies for first four modes are presented in tabular form in Tables 10.4-10.7, respectively. In published work of Ku, 1998, only first two natural frequencies are available. The present study extracts natural frequencies in first four modes. These have been compared with classical closed form solutions. These studies are also conducted by discretising the beam into the three and seven number of elements in order to make convergence comparisons and are summaries in Tables 10.4-10.7 .
